Method For Determining The Integrity Of An Oil Well Plug

ABSTRACT

A method for determining the integrity of a plug plugging an oil well includes the following steps: numerically modelling at least one structural characteristic of the plug as the plug solidifies under depth conditions; detection of the formation of a micro-ring fracture and/or of the deterioration of the plug as a function of the at least one structural characteristic of the plug with respect to a criterion for the formation of a micro-ring fracture and/or rupture; when the formation of a micro-ring fracture is detected, numerically modelling at least one structural characteristic of the plug as the plug solidifies under depth conditions on the basis of the dimensional evolution of the micro-ring fracture over the course of time; and, if the micro-ring fracture extends between two critical points of the well and/or if the plug is damaged, detecting a fault with the plug.

TECHNICAL FIELD

The invention relates to the field of the hydrocarbon extraction industry. It more specifically relates to the operation of hydrocarbon wells, and more particularly, the production of a plug for plugging an oil well.

The invention is particularly advantageously applicable to determining the leak risk of a cement plug of an oil well.

The invention may also be used to estimate the quantity of post-expansion agents necessary in order to limit the leak risk of a cement plug.

PRIOR ART

The configuration of a hydrocarbon well traditionally comprises a set of casings extending between the surface and the reservoir containing the hydrocarbons. The well thus formed can pass through several separate reservoirs, for example a potable water aquifer and a gas pocket.

Closing a hydrocarbon well seeks to reestablish the natural integrity of the formations that have been penetrated by the borehole. More generally, the aim of the closure is to lastingly isolate permeable formations, including those containing hydrocarbons, in order to protect the underground resources, avoid potential contamination of the potable water aquifers and to prevent leaks toward the surface or between reservoirs.

The closure may also be temporary in order to combat losses of circulation or to serve as the basis for the deviation of a well.

To close a well, it is traditional to position cement plugs in line with certain geological formations in order to isolate the reservoirs.

This process is performed by injecting cement in grout form into the well and allowing it to set. The first step consists in preparing the well. In particular, if the cement plug is placed without contact with the bottom of the well and there is a risk of the cement sliding toward the bottom of the well by gravity, a seal is placed below the plug to prevent this descent. This seal may be a viscous or reactive fluid or a mechanical device.

The grout is next prepared on the surface, then pumped such as to fill a part of the drill hole. The grout can be placed in an open hole or in a cased well, i.e., the walls of which are covered with one or several cemented tubular pipes.

The characteristics of the grout are determined based upon a large number of parameters, including the characteristics of the well (density of the drilling fluids, stability of the well, deviation, diameter of the drilled hole), the pumping technique used, whether a seal is used, the location of the cement plug within the well, etc. . . . .

However, during setting, the volume of the grout may decrease due to physical, thermal or chemical processes. There is therefore a risk of a micro-ring being created between the wall and the cement plug. When this micro-ring extends between two separate reservoirs, fluids can migrate along the micro-ring and contaminate one or the other of the two reservoirs. The greatest risk is that of contamination of the potable water aquifers with brine, hydrocarbons or other fluids.

To resolve this problem, it is known to add post-expansion agents to the grout in such a way as to increase the volume of the cement plug after the setting of the grout.

This technique makes it possible to limit the risk of forming a micro-ring. Adding post-expansion agents may however deteriorate the mechanical holding of the cement plug.

To measure the integrity of a cement plug, it is known to perform a pressure test after the setting of the grout. Two types of tests exist.

Pressure increase tests are carried out by closing the wellhead, and increasing the pressure in the well to a set value. The pumps are then stopped and the pressure is measured. During this test, the measured pressure must not decrease at a rate above a predetermined value, for example 35 kPa/min, during a predetermined duration, for example 30 min. If the pressure drop is slower, the test is conclusive. Otherwise, the plug is “leaky” and a solution must be found to offset this.

Pressure decrease tests consist of switching the drilling mud for a lighter drilling mud and following a protocol identical to the previous one, but verifying that the pressure does not increase and not the opposite.

These two tests require use of the well over a relatively long period of time, considering the placement and removal of the devices required in order to perform the tests. But managing the installations of a hydrocarbon well is extremely costly, and may incur costs that may reach a million Euros per day.

As a result, the pressure test is generally performed very early on, just after the cement sets, under conditions that are not optimal, insofar as part of the grout may not be fully hydrated and may deform under pressure without revealing the presence of a micro-ring. Typically, the pressure test is generally performed 12 to 48 hours after pumping the grout into the well. Under these conditions, the post-expansion agents have not yet necessarily begun to take effect.

Furthermore, the pressure test makes it possible to only verify the resistance of the cement plug when the pressure of a fluid is applied to the top of the cement plug, while the cement plug may also experience a pressure variation from below. As a result, there is the risk of a micro-ring forming from the lower part of the cement plug.

Furthermore, the pressure drop test tolerates a pressure drop of about 35 kPa per minute. But the pressure drop does not depend only upon the opening of the micro-ring, but also on the volume of fluid located above the plug, which depends on the length of the plug and the diameter of the well, as well as the compressibility of the fluid. Thus, this pressure drop tolerance is not a representative property of the integrity of the cement plug.

The determination of the characteristics of a grout and the quantity of post-expansion agents is performed in two ways today by using the results of the pressure tests of the previous cement plugs. The pressure tests not being precise, the determination method is however highly imprecise.

The technical problem of the invention consists in determining, with greater precision, whether an oil well plug has a fault, for example a micro-ring extending between two different ordinate points.

BRIEF DESCRIPTION OF THE INVENTION

To resolve this technical problem, the invention proposes to simulate the behavior of the plug during setting over time.

At each incremental moment of the simulation, the formation of a micro-ring is sought. If the conditions are such that no micro-ring can form, the ring meets expectations.

Conversely, if the conditions are met for a micro-ring to form, the evolution of the micro-ring is studied. If the micro-ring extends between two critical points, the plug has a fault.

To that end, the invention relates to a method for determining the integrity of an oil well plug, the method comprising the following steps:

-   -   digitally modeling at least one structural characteristic of the         plug, over the course of time, as the plug solidifies under         depth conditions;     -   detecting, over the course of time, the formation of a         micro-ring fracture and/or the deterioration of the plug as a         function of the at least one structural characteristic of the         plug with respect to a formation and/or rupture criterion;     -   when the formation of a micro-ring fracture is detected,         digitally modeling at least one structural characteristic of the         plug, over the course of time, as the plug solidifies under         depth conditions on the basis of the dimensional evolution of         the micro-ring fracture over the course of time; and     -   if the micro-ring fracture extends between two critical points         of the well and/or if the plug is damaged, detecting a fault         within the plug.

The invention makes it possible to replace the pressure test of a plug with a more precise digital simulation.

Indeed, the digital simulation of the invention makes it possible to estimate the possibilities of the formation of a micro-ring fracture from the upper face and the lower face of the plug.

Furthermore, this digital simulation can study the integrity of the plug over a greater length of time independently of those financial constraints relating to the usage costs of a hydrocarbon well. Furthermore, the invention makes it possible to reduce the usage costs of a hydrocarbon well by eliminating the pressure test.

The invention can be applied to all plug types formed by a solidification reaction, for example plugs made by the setting of a cement.

According to one embodiment, the step for digitally modeling at least one structural characteristic of the plug, over the course of time, as the plug solidifies under depth conditions is performed based upon:

-   -   properties of the plug;     -   geometric characteristics of the plug;     -   properties of the well; and     -   environmental conditions.

The properties of the plug encompass the physical, chemical, mechanical, thermal and hydraulic properties. The properties of the well for example incorporate the permeability of the formations, the rigidity of the metal tube, etc. The environmental conditions make it possible to determine whether the setting reaction is performed with or without added water.

This embodiment makes it possible to simulate, with great precision, the solidification of the grout under basic conditions over the course of time.

According to one embodiment, the step for digitally modeling at least one structural characteristic of the plug, over the course of time, as the plug solidifies under depth conditions takes account of the evolution of the stresses experienced by the plug, the evolution of the deformation of the plug, the evolution of the chemical setting reaction of the plug, the evolution of the temperature at the plug during setting, the evolution of the properties of the plug and the evolution of the pressure of the pores of the plug.

This embodiment also makes it possible to simulate, with very great precision, the solidification of the grout under bottom conditions over the course of time.

According to one embodiment, the method also comprises the following additional steps:

-   -   determining dimensions of the micro-ring fracture; and     -   determining the quantity of post-expansion agents necessary to         increase the volume of the plug in order to fill in the         determined dimensions of the micro-ring fracture.

This embodiment makes it possible to dimension the necessary quantity of post-expansion agents precisely.

Advantageously, the damage to the plug is detected in traction or shear, for example using the Mohr-Coulomb criterion.

This embodiment makes it possible to detect a failure of the plug, for example when the quantity of post-expansion agents degrades the structural characteristics.

According to one embodiment, during the step for digitally modeling at least one structural characteristic of the plug, over the course of time, as the plug solidifies under depth conditions, the method also comprises the following steps:

-   -   detecting an end of setting corresponding to an evolution of the         degree of solidification per hour of the plug below 2·10⁻⁴;     -   determining a pressure applied above the plug from fluids placed         above the plug after detecting an end of setting;     -   determining a pressure applied below the plug from fluids placed         below the plug after detecting an end of setting; and     -   determining the stresses experienced by the plug, after setting,         based upon pressures applied above and below the plug.

This embodiment makes it possible to estimate the stresses experienced by the plug after the setting of the grout, based upon fluids present within the well.

According to one embodiment, during the step consisting of determining the stresses experienced by the plug, after setting, as a function of the pressures applied above and below the plug, the method also comprises the following additional steps:

-   -   simulating the propagation of a chance micro-ring fracture that         appears above or below the plug, as a function of the pressures         applied above and below the plug;     -   if the chance micro-ring fracture extends between two critical         points, determining a flow rate of the chance micro-ring         fracture; and     -   if the flow rate of the chance micro-ring fracture is above a         threshold, detecting a fault with the plug.

This embodiment makes it possible to estimate the evolution of a micro-ring fracture that is formed after random events, for example the compacting of a reservoir.

According to one embodiment, after the step consisting of detecting a fault on the plug by the presence of a chance micro-ring fracture, the method also comprises the following steps:

-   -   determining the dimension of the chance micro-ring fracture; and     -   determining the quantity of post-expansion agents necessary to         increase the volume of the plug in order to fill in the         determined dimension of the chance micro-ring fracture.

This embodiment makes it possible to dimension the necessary quantity of post-expansion agents precisely in order to avoid the propagation of a chance micro-ring fracture.

BRIEF DESCRIPTION OF THE FIGURES

The way to implement the invention as well as the advantages deriving therefrom will be clearly seen from the description of the following embodiment, supported by the appended figures in which:

FIG. 1 is a sectional view of a section of an oil well closed by a cement plug;

FIG. 2 is a flowchart of the steps for determining the integrity of a cement plug of an oil well;

FIG. 3 is a sectional view of one end of a cement plug having a micro-ring fracture;

FIG. 4 is an illustration of the evolution of a micro-ring fracture as a function of the thermal-chemical-poroelastic properties of the cement plug.

WAYS TO IMPLEMENT THE INVENTION

FIG. 1 illustrates an oil well 10 with a circular section comprising several separate reservoirs 15-17. The reservoirs 15-17 are separated by tight geological barriers 14. For example, the first reservoir 15 may correspond to a potable water aquifer, the second reservoir 16 may correspond to a gas pocket and the third reservoir 17 may correspond to an oil reservoir.

The invention aims at the formation of a plug 12, for example in order to lastingly isolate the reservoirs 15-17. In the example of FIG. 1, the cement plug 12 is formed in a casing 11 made by a metal tube, which in turn is cemented to the walls of the well 10. Alternatively, the plug 12 can be made directly on the walls of the well 10. The well 10 can also have several casings 11 and several plugs 12 without changing the invention.

Furthermore, the embodiment of FIGS. 1 to 4 describes a cement plug 12. Alternatively, the plug can be made from a material other than cement, for example a resin.

The cement plug 12 must guarantee isolation between several critical points A-D. For example, the cement plug 12 seeks to prevent contamination of the gas pocket from the oil reservoir. To that end, no fluid must circulate between the critical points C and D or between a lower end 21 of the cement plug 12 and the critical point C. For another example, the cement plug 12 seeks to guarantee the sealing of the well 10 with the potable water aquifer. To that end, no fluid must circulate between an upper end 20 of the cement plug 12 and the potable water aquifer.

In the example of FIG. 1, the cement plug 12 is faulty because, in the case where the metal wall of the tube were to disappear due to corrosion, it allows for contamination of the potable water aquifer by the gas pocket due to the presence of a micro-ring fracture 25 extending between the critical points A and B.

Thus, there are therefore many possibilities for the failure of a cement plug 12. To determine whether a cement plug 12 is effective, FIG. 2 illustrates a method for determining the integrity of a cement plug 12 according to one embodiment of the invention.

The first step consists of modeling 100, over the course of time t, at least one structural characteristic of the cement plug 12 during the hydration ξ of the cement plug 12 under depth conditions.

Preferably, this step 100 is carried out as a function of:

-   -   properties of the plug 12;     -   geometric characteristics of the plug 12;     -   properties of the well 10; and     -   environmental conditions.

In the continuation of the description, an example embodiment will be precisely described. To describe the hydration of the cement plug 12, the cement plug 12 is considered in a cylindrical coordinate system rθz as illustrated along the plane rz of FIG. 1. The cement plug 12 can be defined, as a function of the hydration ζ thereof varying between 0 and 1, according to the following characteristics:

-   -   the linear thermal expansion coefficient α;     -   the friction angle φ, generally measured using triaxial tests;     -   Poisson's ratio u;     -   Skempton's coefficient B;     -   the tensile strength BS, obtained at the interface between the         cement plug 12 and the well 10;     -   the Young's modulus E;     -   the shear modulus G;     -   the incompressibility modulus K;     -   the Biot's modulus when the drilling mud forming the cement plug         12 is saturated with liquid M;     -   the tensile strength TS, obtained by measuring the tensile         strength of the cement alone;     -   the uniaxial compression strength UCS;     -   Biot's coefficient b; and     -   the hydration coefficient s.

Furthermore, the fluid located above or below the plug (12) has the following characteristics:

-   -   the plastic threshold of the fluid within the meaning of the         Herschel-Bulkley definition τ;     -   the consistency of the fluid within the meaning of the         Herschel-Bulkley definition k;     -   the index of the fluid within the meaning of the         Herschel-Bulkley definition n.

Not all of these characteristics are necessarily necessary in order to model the hydration of a cement plug 12 as a function of the precision of the model used.

These characteristics are described in the following works:

-   Coussy, O.: “Mécanique des Milieux Poreux”, Editions Technip (1991)     437; -   Charlez, Ph. A.: “Rock Mechanics, Volume 1. Theoretical     Fundamentals”, Editions Technip (1991) 333; and -   Wang, H. F.: “Theory of Linear Poroelasticity with Applications to     Geomechanics and Hydrogeology”, Princeton University Press (2000)     287.

Certain features vary depending upon whether the environmental conditions resulted in a drained or non-drained behavior of the cement 12. For the continuation of the development, these features will take the index k for a development that may occur independently under the drained or non-drained conditions, the index d when they will be considered under the drained conditions and the index u when they will be considered under the non-drained conditions. Furthermore, a characteristic h_(k) will assume the value 0 under the drained conditions and the value 1 under the non-drained conditions.

Furthermore, these characteristics depend upon one another according to the following equations:

${v_{d} = \frac{{3v_{u}} - {\left( {1 + v_{u}} \right) \cdot {bB}}}{3 - {2{\left( {1 + v_{u}} \right) \cdot {bB}}}}};{v_{u} = \frac{{3v_{d}} + {\left( {1 - {2v_{d}}} \right) \cdot {bB}}}{3 - {\left( {1 - {2v_{d}}} \right) \cdot {bB}}}};$ ${b = {\frac{1}{B} \cdot \left( {1 - \frac{K_{d}}{K_{u}}} \right)}};{B = {3\; \frac{v_{u} - v_{d}}{\left( {1 - {2v_{d}}} \right) \cdot \left( {1 + v_{u}} \right) \cdot b}}};$ ${B = \frac{K_{u} - K_{d}}{b \cdot K_{u}}};{E_{d} = {\frac{3\left( {1 - {bB}} \right)}{3 - {2{\left( {1 + v_{u}} \right) \cdot {bB}}}} \cdot E_{u}}};$ ${E_{u} = \frac{3E_{d}}{3 - {\left( {1 - {2v_{d}}} \right) \cdot {bB}}}};{G = \frac{E_{k}}{2\left( {1 + v_{k}} \right)}};{K_{k} = \frac{E_{k}}{3\left( {1 - {2v_{k}}} \right)}};$ ${K_{d} = {K_{u} - {b^{2} \cdot M}}};{K_{u} = \frac{K_{d}}{1 - {bB}}};{M = {\frac{B \cdot K_{u}}{b}.}}$

The hydration of the cement plug 12 may be modeled as a function of the following elements:

-   -   the applied stresses σ;     -   the deformation ε,     -   the pressure of the pores P_(p);     -   the vaporization pressure of the water in the pores of the         cement P_(vapor);     -   the radial displacement U_(r);     -   the temperature of the reaction T;     -   the degree of advancement of the hydration of the cement or the         entire chemical reaction ξ; and     -   the imposed pressure P_(pores) similar to a transfer of mass         from the outside of the cement onto the cement, for example the         addition of water.

For example, the stresses σ and pore pressures P_(p) experienced by the cement plug 12 over the course of time t during the hydration of the cement plug 12 under the depth conditions are modeled by the following equations when the cement is in a saturated environment:

$\underset{\underset{\_}{\_}}{\delta \; \sigma} = {{{\left( {K_{d} - \frac{2G}{3}} \right) \cdot {tr}}\; {\underset{\underset{\_}{\_}}{\delta \; ɛ} \cdot \underset{\underset{\_}{\_}}{I}}} + {2{G \cdot \underset{\underset{\_}{\_}}{\delta \; ɛ}}} + {{b \cdot \delta}\; {p_{P} \cdot \underset{\underset{\_}{\_}}{I}}} + {3{K_{d} \cdot \left( {{{s_{d} \cdot \delta}\; \xi} + {{\alpha_{d} \cdot \delta}\; T}} \right) \cdot \underset{\underset{\_}{\_}}{I}}}}$ ${\delta \; p_{P}} = {{M \cdot \left( {{{b \cdot \; {tr}}\; \underset{\underset{\_}{\_}}{\delta \; ɛ}} + \frac{\delta \; m}{\rho_{l}}} \right)} + {3\; {\frac{{K_{u} \cdot \alpha_{u}} - {K_{d} \cdot \alpha_{d}}}{b} \cdot \delta}\; T} - {3\; {\frac{{{K_{u} \cdot s_{u}} + {K_{d} \cdot s_{d}}}\;}{b} \cdot \delta}\; \xi}}$

This system of equations takes into account the evolution of the deformation ε of the cement plug 12 by means of Hooke's law. Furthermore, this system of equations models the evolution of the chemical setting reaction ξ of the cement plug 12 and the evolution of the temperature T at the cement plug 12 during setting, since the chemical reaction is heat-activated. In conclusion, this system of equations is a system with two coupled equations wherein the evolution of the stresses σ is a function of the evolution of the pore pressures P_(p).

In order to simplify the solving of this system, the characteristics of the cement plug 12 can be integrated into the following coefficients:

${C_{1}^{(k)} = {K_{k} - \frac{2G}{3}}};{C_{2}^{(k)} = {3{K_{k} \cdot \alpha_{k}}}};{C_{3}^{(k)} = {3{\left( {1 - {2h_{k}}} \right) \cdot K_{k} \cdot s_{k}}}};$ ${C_{4}^{(k)} = {h_{k} \cdot b}};{C_{5}^{(k)} = {h_{k} \cdot M \cdot b}};{C_{6}^{(k)} = {3{h_{k} \cdot \frac{{K_{u} \cdot \alpha_{u}} - {K_{d} \cdot \alpha_{d}}}{b}}}};$ ${C_{7}^{(k)} = {{{- 3}{h_{k} \cdot \frac{{K_{u} \cdot s_{u}} + {K_{d} \cdot s_{d}}}{b}}} = {{- h_{k}} \cdot M \cdot \frac{m_{\xi}}{\rho_{l}}}}};{C_{8}^{(k)} = h_{k}};$ $C_{9}^{(k)} = {K_{k} + \frac{4G}{3}}$

To solve the system, it is possible to emit hypotheses concerning the behavior of the cement as a function of the environment of the well 10 and the structure of the cement. For example, it is possible to consider that the cement is drained. It follows that the pressure increments of the pores can dissipate naturally and the pore pressure increments P_(p) are nil.

A non-drained behavior is modeled by assuming the absence of water transfer, i.e., with δm=0. By using the preceding coefficients, the stresses σ and pore pressures P_(p) experienced by the cement plug 12 can be written as follows:

δσ=C ₁ ^((u)) ·trδε·I+2G·δε+C ₂ ^((u)) ·δT·I+C ₃ ^((u)) ·δξ·I

δp _(p) =C ₅ ^((u)) ·trδε+C ₆ ^((u)) ·δT+C ₇ ^((u))·δξ

According to another hypothesis, the water can be removed from the cement, which results in an increment in the pore pressures P_(pores). According to this hypothesis, the stresses σ and pore pressures P_(p) experienced by the cement plug 12 can be written as follows:

δσ=C ₁ ^((k)) ·trδε·I+2G·δε+C ₂ ^((k)) ·δT·I+C ₃ ^((k)) ·δξ·I+C ₄ ^((k)) ·δp _(pores) ·I

δp _(p) =C ₅ ^((k)) ·trδε+C ₆ ^((k)) ·δT+C ₇ ^((k)) ·δξ+C ₈ ^((k)) ·δp _(pores)

In the following paragraphs, the solving of the stress σ and pore pressure P_(p) equations experienced by the cement plug 12 will be presented in the case of the hydration of a saturated cement plug 12, considering that the deformations experienced are planar.

To that end, the following changes in variables can be used:

δ Υ₁^((k)) = δ Υ₄^((k)) + C₄^((k)) ⋅ δ p_(pores); δΥ₂^((k)) = δ Υ₅^((k)) + C₈^((k)) ⋅ δ p_(pores); δ Υ₃^((k)(0)) = δ Υ₆^((k)(0)) + C₄^((k)(0)) ⋅ δ p_(pores)⁽⁰⁾; δ Υ₄^((k)) = C₂^((k)) ⋅ δ T + C₃^((k)) ⋅ δ ξ; δ Υ₅^((k)) = C₆^((k)) ⋅ δ T + C₇^((k)) ⋅ δ ξ; δ Υ₆^((k)(0)) = (C₂^((k)(0)) + k₁) ⋅ δ T⁽⁰⁾ + C₃^((k)(0)) ⋅ δ ξ⁽⁰⁾; ${{\delta \; \mathrm{\Upsilon}_{7}} = {{\delta \; \mathrm{\Upsilon}_{4}^{(d)}} + {C_{4}^{(u)} \cdot \frac{\int_{0}^{r_{0}}{\frac{\rho_{l}}{M}\delta \; {\mathrm{\Upsilon}_{5}^{(u)} \cdot r \cdot {dr}}}}{\int_{0}^{r_{0}}{\frac{\rho_{l}}{M} \cdot r \cdot {dr}}}}}};$ ${{\delta \; \mathrm{\Upsilon}_{8}^{(0)}} = {{\delta \; \mathrm{\Upsilon}_{6}^{{(d)}{(0)}}} + {C_{4}^{{(u)}{(0)}} \cdot \frac{\int_{0}^{r_{0}}{{\frac{\rho_{l}}{M} \cdot \delta}\; {\mathrm{\Upsilon}_{5}^{(u)} \cdot r \cdot {dr}}}}{\int_{0}^{r_{0}}{\frac{\rho_{l}}{M} \cdot r \cdot {dr}}}}}};$ δ Υ₉^((k)) = C₄^((k)) ⋅ δ p_(P); δ Υ₁₀ = δ Υ₄^((d)) + C₄^((u)) ⋅ δ p_(vapor); δ Υ₁₁⁽⁰⁾ = δ Υ₆^((d)(0)) + C₄^((u)(0)) ⋅ δ p_(vapor)⁽⁰⁾; δΥ₁₂^((k)) = (C₂^((k)) − C₆^((k))) ⋅ δ T + (C₃^((k)) − C₇^((k))) ⋅ δ ξ; δΥ₁₃^((k)) = δ Υ₁₂^((k)) + (C₄^((k)) − C₈^((k))) ⋅ δ p_(pores); ${{\delta\mathrm{\Upsilon}}_{14} = {{\delta \; \mathrm{\Upsilon}_{4}^{(d)}} + {\left( {C_{4}^{(u)} - C_{8}^{(u)}} \right) \cdot \frac{\int_{0}^{r_{0}}{{\frac{\rho_{l}}{M} \cdot \delta}\; {\mathrm{\Upsilon}_{5}^{(u)} \cdot r \cdot {dr}}}}{\int_{0}^{r_{0}}{\frac{\rho_{l}}{M} \cdot r \cdot {dr}}}}}};$ δ Υ₁₅ = δ Υ₄^((d)) + (C₄^((u)) − C₈^((u))) ⋅ δ p_(p); ${\Psi_{1}^{(k)} = {C_{4}^{(k)} \cdot \left\lbrack {\frac{\int_{0}^{r_{0}}{\frac{\rho_{l}}{M} \cdot {p_{p}(\xi)} \cdot r \cdot {dr}}}{{\int_{0}^{r_{0}}{\frac{\rho_{l}}{M} \cdot r \cdot {dr}}}\;} - {p_{p}(\xi)}} \right\rbrack}};{\Psi_{2} = {{\sigma_{r}(\xi)} - {p_{p}(\xi)}}};$ ${\Psi_{3}^{(k)} = {\Psi_{2} + {\left( {C_{4}^{(k)} - C_{8}^{(k)}} \right) \cdot \left( {\frac{\int_{0}^{r_{0}}{\frac{\rho_{l}}{M} \cdot {p_{p}(\xi)} \cdot r \cdot {dr}}}{\int_{0}^{r_{0}}{\frac{\rho_{l}}{M} \cdot r \cdot {dr}}} - {p_{p}(\xi)}} \right)}}};$ Ψ₄ = σ_(r)(ξ) − p_(vapor)

wherein the indices r, θ and z are used to designate the normal components of straining mechanisms (for example, the straining mechanism of the stresses) in the radial, orthoradial and vertical directions. They are also used to designate the components of the vectors (for example, the vector of the displacements) in the same directions. The exponent 0 is used to designate the variables calculated at the interface between the plug 12 and the well 10, for a radius equal to r0.

Considering a cylindrical plug 12 with radius r₀ made up of a thermal-chemical-poroelastic material experiencing pressure increments at the periphery thereof, with temperature δT, hydration δξ, imposed pore pressure δp_(pores), the equations modeling the stresses σ and the pore pressures P_(p) experienced by the cement plug 12 can be written as follows:

δσ_(r) =C ₉ ^((k))·δε_(r) +C ₁ ^((k))·(δε_(θ)+δε_(z))+δγ₁ ^((k))

δσ_(θ) =C ₉ ^((k))·δε_(θ) +C ₁ ^((k))·(δε_(r)+δε_(z))+δγ₁ ^((k))

δσ_(z) =C ₉ ^((k))·δε_(z) +C ₁ ^((k))·(δε_(r)+δε_(θ))+δγ₁ ^((k))

δp _(p) =C ₅ ^((k))·(δε_(r)+δε_(θ)+δε_(z))+δγ₂ ^((k))

In assuming a position far from the ends 20, 21 of the plug 12, i.e., under planar deformation conditions, the equations can be written as follows:

δσ_(r) =C ₉ ^((k))·δε_(r) +C ₁ ^((k))·δε_(θ)+δγ₁ ^((k))

δσ_(θ) =C ₉ ^((k))·δε_(θ) +C ₁ ^((k))·δε_(r)+δγ₁ ^((k))

δσ_(z) =C ₉ ^((k))·(δε_(r)+δε_(θ))+δγ₁ ^((k))

δp _(p) =C ₅ ^((k))·(δε_(r)+δε_(θ))+δγ₂ ^((k))

The equilibrium equation can be written in the following form:

${\frac{{\partial\delta}\; \sigma_{r}}{\partial r} + \frac{{\delta \; \sigma_{r}} - {\delta \; \sigma_{\theta}}}{r}} = 0$

The global equation is written in the following form:

${{2{G \cdot \frac{{\delta \; ɛ_{r}} - {\delta ɛ}_{\theta}}{r}}} + {\frac{\partial}{\partial r}\left( {{{C_{9}^{(k)} \cdot \delta}\; ɛ_{r}} + {{C_{1}^{(k)} \cdot \delta}\; ɛ_{\theta}}} \right)} + \frac{{\partial\delta}\; \mathrm{\Upsilon}_{1}^{(k)}}{\partial r}} = 0$

Furthermore, the equations linking the deformations and the displacements are written in symmetry of revolution in the following form:

${\delta ɛ}_{r} = \frac{{\partial\delta}\; u_{r}}{\partial r}$ ${\delta \; ɛ_{\theta}} = \frac{\delta \; u_{r}}{r}$

The global equation can therefore be rewritten in the following form:

${{\frac{2G}{r} \cdot \left( {\frac{{\partial\delta}\; u_{r}}{\partial r} - \frac{\delta \; u_{r}}{r}}\; \right)} + {\frac{\partial}{\partial r}\left( {{C_{9}^{(k)} \cdot \frac{{\partial\delta}\; u_{r}}{\partial r}} + {C_{1}^{(k)} \cdot \frac{\delta \; u_{r}}{r}}} \right)} + \frac{\partial{\delta\mathrm{\Upsilon}}_{1}^{(k)}}{\partial r}} = 0$

After several manipulations, this equation can be rewritten in the following form:

${{C_{9}^{(k)} \cdot \frac{{\partial^{2}\delta}\; u_{r}}{\partial r^{2}}} + {\left( {\frac{\partial C_{9}^{(k)}}{\partial r} + \frac{C_{9}^{(k)}}{r}} \right) \cdot \frac{{\partial\delta}\; u_{r}}{\partial r}} + {\left( {\frac{\partial C_{1}^{(k)}}{\partial r} - \frac{C_{9}^{(k)}}{r}} \right) \cdot \frac{\delta \; u_{r}}{r}} + \frac{{\partial\delta}\; \mathrm{\Upsilon}_{1}^{(k)}}{\partial r}} = 0$

This equation is solved taking two limit conditions into account.

The first limit condition defines the interaction between the cement plug 12 and the well 10 wherein it is placed. Assuming a linear behavior for the displacement at the wall of the well 10, this limit condition can be written in the following form:

${\delta \; \sigma_{r}^{(0)}} = {{{- k_{0}} \cdot \frac{\delta \; u_{r}^{(0)}}{r_{0}}} - {{k_{1} \cdot \delta}\; T^{(0)}}}$

wherein the values k₀ and k₁ depend upon the properties of the materials of the well 10 and the geometry of the well 10, such as the diameter of the hole and the inner and outer diameters of the metal tube.

The second limit condition describes, for symmetry reasons, the nullity of the displacement at the center of the cement plug 12:

δu _(r)|_(r=0)=0

It follows that the first limit condition can be rewritten as follows:

${C_{9}^{{(k)}{(0)}} \cdot \frac{{\partial\delta}\; u_{r}}{\partial r}}{_{r = r_{0}}{{{{+ \left( {C_{1}^{{(k)}{(0)}} + k_{0}} \right)} \cdot \frac{\delta \; u_{r}^{(0)}}{r_{0}}} + {\delta \; \mathrm{\Upsilon}_{3}^{{(k)}{(0)}}}} = 0}}$

The system of equations to be solved is therefore written as follows:

${{{{{{C_{9}^{(k)} \cdot \frac{{\partial^{2}\delta}\; u_{r}}{\partial r^{2}}} + {\left( {\frac{\partial C_{9}^{(k)}}{\partial r} + \frac{C_{9}^{(k)}}{r}} \right) \cdot \frac{{\partial\delta}\; u_{r}}{\partial r}} + {\left( {\frac{\partial C_{1}^{(k)}}{\partial r} - \frac{C_{9}^{(k)}}{r}} \right) \cdot \frac{\delta \; u_{r}}{r}} + \frac{{\partial\delta}\; \mathrm{\Upsilon}_{1}^{(k)}}{\partial r}} = 0}\mspace{20mu} {C_{9}^{{(k)}{(0)}} \cdot \frac{{\partial\delta}\; u_{r}}{\partial r}}}}_{r = r_{0}} + {\left( {C_{1}^{{(k)}{(0)}} + k_{0}} \right) \cdot \frac{\delta \; u_{r}^{(0)}}{r_{0}}} + {\delta \; \mathrm{\Upsilon}_{3}^{{(k)}{(0)}}}} = 0$   δ u_(r|r = 0) = 0

It is thus possible to solve this system of equations and to obtain the displacements of the cement plug 12 over the course of time t.

To that end, it is possible to consider a final hypothesis wherein the cement is impermeable, i.e., the system is not drained and the increment of the pore pressures δp_(pores) is nil. According to another possible hypothesis, the cement is placed opposite a permeable formation, i.e., the system is drained. According to another possible hypothesis, the cement is permeable without adding water. According to another hypothesis, the cement undergoes a variation in pore pressures. In conclusion, the equations linking the deformations and the displacements make it possible to obtain the stresses σ and the pore pressures P_(p) experienced by the cement plug 12. Alternatively, other structural characteristics of the cement can be modeled.

Solving the system of equations therefore makes it possible to obtain the stress evolution σ and the pore pressures P_(p) experienced by the cement plug 12 over the course of time t during hydration ξ under depth conditions.

When the stresses σ and/or the pore pressures P_(p) experienced by the cement plug 12 exceed a threshold value, the cement plug 12 is favorable to the formation of a micro-ring fracture 25. The second step of the method therefore consists in detecting 101, over the course of time t, whether the stresses σ and the pore pressures P_(p) experienced by the cement plug 12 exceed the threshold value, in order to adapt the digital model in integrating the formation of a micro-ring fracture 25.

When the formation of a micro-ring fracture 25 is detected, the method reaches a step 102 wherein the modeling is performed by taking into account the dimensional evolution of the micro-ring fracture 25 over the course of time. For example, it is possible to consider that the model with no micro-ring fracture 25 is valid as long as the tensile strength of the interface between the plug and the well 10 is not exceeded, which is reflected by the following equation:

σ_(r) ⁽⁰⁾ −P _(p) ⁽⁰⁾ >−BS

Furthermore, the opening of the micro-ring fracture 25 is detected when the preceding equation is not verified. This opening of the micro-ring fracture 25 can be characterized by the following equation:

${{micro}\text{-}{ring}\mspace{14mu} {fracture}} = {{\Delta \; u_{r}^{(0)}} + {r_{0} \cdot \left( {{{\frac{1}{k_{0}} \cdot \Delta}\; \sigma_{r}^{(0)}} + {{\frac{k_{1}}{k_{0}} \cdot \Delta}\; T^{(0)}}} \right)}}$

wherein Δσ_(r) ⁽⁰⁾, Δu_(r) ⁽⁰⁾, and ΔT⁽⁰⁾ are the stress, displacement and temperature variations at the periphery of the plug calculated from the beginning of the hydration of the cement 12. The first two parameters are calculated in the following paragraphs.

The hypothesis is adopted that the micro-ring fracture 25 does not reach the upper 20 or lower 21 limits of the cement plug 12, the pore pressure in the cement is greater than the vaporization pressure of the water in the pores and the behavior is not drained. The radial stress at the interface between the plug and the well 10 is equal to the sum of the calculated stress for a degree of hydration ξ and the stress increment due to the increase of the degree of hydration δξ according to the following formula:

σ_(r) ⁽⁰⁾=σ_(r) ⁽⁰⁾(ξ)+δσ_(r) ⁽⁰⁾

In this example, the pore pressure at the interface between the plug and the well 10 is equal to the sum of the calculated pressure for a degree of hydration ξ and the pressure increment due to the increase of the degree of hydration δξ according to the following formula:

p _(p) ⁽⁰⁾ =p _(p) ⁽⁰⁾(ξ)+δp _(p) ⁽⁰⁾

The radial stress at the interface between the plug and the well 10 is equal to the pore pressure at said interface according to the following formula:

δσ_(r) ⁽⁰⁾ −δp _(p) ⁽⁰⁾ =p _(p) ⁽⁰⁾(ξ)−σ_(r) ⁽⁰⁾(ξ)

The radial stress increment at the interface between the plug and the well 10 is then calculated according to the thermal-chemical-poroelastic model under non-drained conditions according to the following relationship:

δσ_(r) ⁽⁰⁾ =C ₉ ^((u)(0))·δε_(r) ⁽⁰⁾ +C ₁ ^((u)(0))·δε_(θ) ⁽⁰⁾+δγ₄ ^((u)(0))

As a result, the pore pressure increment can be written as follows:

δp _(p) ⁽⁰⁾ =C ₅ ^((u)(0))·(δε_(r) ⁽⁰⁾+δε_(θ) ⁽⁰⁾)+δγ₅ ^((u)(0))

By using the changes in variables, the equation can be written as follows:

(C ₉ ^((u)(0)) −C ₅ ^((u)(0)))·δε_(r) ⁽⁰⁾+(C ₁ ^((u)(0)) −C ₅ ^((u)(0)))·δε_(θ) ⁽⁰⁾+δγ₁₂ ^((u)(0))+Ψ₂ ⁽⁰⁾=0

By using the equations linking the deformations and the displacements, the equation can be written in the following form:

${{{\left( {C_{9}^{{(u)}{(0)}} - C_{5}^{{(u)}{(0)}}} \right) \cdot \frac{\delta \; d\; u_{r}}{\partial r}}}_{r = r_{0}} + {\left( {C_{1}^{{(u)}{(0)}} - C_{5}^{{(u)}{(0)}}} \right) \cdot \frac{\delta \; u_{r}^{(0)}}{r_{0}}} + {\delta \; \mathrm{\Upsilon}_{12}^{{(u)}{(0)}}} + \Psi_{2}^{(0)}} = 0$

Thus, the system of equations to be solved is as follows:

${{{{{{C_{9}^{(u)} \cdot \frac{{\partial^{2}\delta}\; U_{r}}{\partial r^{2}}} + {\left( {\frac{\partial C_{9}^{(u)}}{\partial r} + \frac{C_{9}^{(u)}}{\partial r}} \right) \cdot \frac{{\partial\delta}\; U_{r}}{\partial r}} + {\left( {\frac{\partial C_{1}^{(u)}}{\partial r} - \frac{C_{9}^{(u)}}{r}} \right) \cdot \frac{\partial u_{r}}{r}} + \frac{{\partial\delta}\; \mathrm{\Upsilon}_{4}^{(u)}}{\partial r}} = 0}{\left( {C_{9}^{{(u)}{(0)}} - C_{5}^{{(u)}{(0)}}} \right) \cdot \left( \frac{{\partial d}\; u_{r}}{\partial r} \right)}}}_{r = r_{0}} + {\left( {C_{1}^{{(u)}{(0)}} - C_{5}^{{(u)}{(0)}}} \right) \cdot \frac{\partial u_{r}^{(0)}}{r_{0}}} + {\delta \; \mathrm{\Upsilon}_{12}^{{(u)}{(0)}}} + \Psi_{2}^{(0)}} = 0$   δ u_(r)_(r = 0) = 0

In the same manner as before, it is thus possible to solve this system of equations and to obtain the displacements of the cement plug 12 over the course of time t.

It follows that it is possible to obtain the stresses σ and the pore pressures P_(p) experienced by the cement plug 12. Furthermore, this model makes it possible to obtain the evolution of the micro-ring fracture 25 over the course of time t.

Considering the micro-ling fracture 25 illustrated in FIG. 3, the model makes it possible to obtain the widths of the micro-ring fracture 25 at several heights e1 and e2. Furthermore, over the course of time t, the model makes it possible to determine the geometric evolutions t1-t4 of the micro-ring fracture 25 as a function of the evolution of the heights e1 and e2 and the characteristics of the cement. As an example, these geometric evolutions t1 to t4 are illustrated in FIG. 4 for different cements.

The geometric evolutions t1 to t3 extend from the upper face 20 of the cement plug 12 to the casing 11 at the geological barrier 14. Thus, although a micro-ring fracture 25 forms in the cement plug 12, it is not dangerous because it does not evolve between two critical points A-D. Furthermore, the geometric evolution t4 extends between the upper face 20 and the potable water aquifer 15. There is therefore a risk of leakage of the cement plug 12. Step 103 seeks to detect whether a micro-ring fracture 25 extends between two critical points A-D.

If no micro-ring fracture 25 extends between two critical points A-D, the modeling continues. Otherwise, a fault of the plug is detected in step 104. The fault is then characterized by determining 105 the dimensions of the micro-ring fracture 25. To that end, the model can be used to simulate the behavior of the cement plug while a micro-ring fracture extends between two critical points A-D.

For example, if a micro-ring fracture 25 reaches the top of the cement plug 12, it follows that the axial forces are no longer limited by the friction of the cement plug 12 on the walls of the well 10: the planar deformation condition no longer applies. It is therefore necessary to add an equal axial stress according to the following formula:

${\delta \; P_{axial}} = {P_{mud} + P_{bouchon} - {\frac{2}{r_{0}^{2}} \cdot {\int_{0}^{r_{0}}{{\sigma_{z}^{({hyd})}(\xi)} \cdot r \cdot {dr}}}}}$

wherein:

P_(mud) is the drilling mud pressure applied to the top of the cement plug;

P_(plug) is the stress due to the weight of the plug located above the considered point; and

σ_(z) ^((hyd))(ξ) is the axial stress in the plug before the micro-ring fracture reaches the top of the cement plug.

Adopting the hypothesis that the micro-ring fracture 25 is so fine that there is no pressure diffused therein. The calculation of the deformations and stresses induced by this axial pressure can be written as follows:

δσ_(r) =C ₉ ^((k))·δε_(r) +C ₁ ^((k))·(δε_(θ)+δε_(z))

δσ_(θ) =C ₉ ^((k))·δε_(θ) +C ₁ ^((k))·(δε_(r)+δε_(z))

δσ_(z) =C ₉ ^((k))·δε_(z) +C ₁ ^((k))·(δε_(r)+δε_(θ))

δp _(p) =C ₅ ^((k))·(δε_(r)+δε_(θ)+δε_(z))

After several manipulations similar to those performed previously, the equation can be written as follows:

${{C_{9}^{(k)} \cdot \frac{{\partial^{2}\delta}\; u_{r}}{\partial r^{2}}} + {\left( {\frac{\partial C_{9}^{(k)}}{\partial r} + \frac{C_{9}^{(k)}}{r}} \right) \cdot \frac{{\partial\delta}\; u_{r}}{\partial r}} + {\left( {\frac{\partial C_{1}^{(k)}}{\partial r} - \frac{C_{9}^{(k)}}{r}} \right) \cdot \frac{\delta \; u_{r}}{r}} + {{\frac{\partial C_{1}^{(k)}}{\partial r} \cdot \Delta}\; ɛ_{z}}} = 0$

Three limit conditions are necessary to solve this equation. The first defines the nullity of the stress variations at the interface between the cement plug 12 and the well 10 according to the following equation:

δσ_(r) ⁽⁰⁾=0

The second describes the nullity of the displacement at the center of the plug, for reasons of symmetry according to the following equation:

δu _(r)|_(r=0)=0

The third describes the axial equilibrium of the cement plug 12 according to the following equation:

2 ⋅ ∫₀^(r₀)δ σ_(z) ⋅ r ⋅ dr = r₀² ⋅ δ P_(axial)

These three limit conditions make it possible to obtain the following system of equations:

${{{{{{C_{9}^{(k)} \cdot \frac{{\partial^{2}\delta}\; u_{r}}{\partial r^{2}}} + {\left( {\frac{\partial C_{9}^{(k)}}{\partial r} + \frac{C_{9}^{(k)}}{r}} \right) \cdot \frac{{\partial\delta}\; u_{r}}{\partial r}} + {\left( {\frac{\partial C_{1}^{(k)}}{\partial r} - \frac{C_{9}^{(k)}}{r}} \right) \cdot \frac{\delta \; u_{r}}{r}} + {{\frac{\partial C_{1}^{(k)}}{\partial r} \cdot \delta}\; ɛ_{z}}} = 0}\mspace{20mu} {C_{9}^{{(k)}{(0)}} \cdot \frac{{\partial\delta}\; u_{r}}{\partial r}}}}_{r = r_{0}} + {C_{1}^{{(k)}{(0)}} \cdot \frac{\delta \; u_{r}^{(0)}}{r_{0}}} + {{C_{1}^{{(k)}{(0)}} \cdot \delta}\; ɛ_{z}}} = 0$ $\mspace{20mu} {{\delta \; ɛ_{z}} = \frac{\begin{matrix} {{r_{0}^{2} \cdot \left( {P_{mud} + P_{plug}} \right)} - {2{\int_{0}^{r_{0}}{{\sigma_{z}^{({hyd})}(\xi)} \cdot r \cdot {dr}}}} -} \\ {2 \cdot {\int_{0}^{r_{0}}{C_{1}^{(k)} \cdot \left( {{\frac{{\partial\delta}\; u_{r}}{\partial r} \cdot r} + {\delta \; u_{r}}} \right) \cdot {dr}}}} \end{matrix}}{2 \cdot {\int_{0}^{r_{0}}{C_{9}^{(k)} \cdot r \cdot {dr}}}}}$   δ u_(t|t = 0) = 0

In the same manner as before, it is thus possible to solve this system of equations and to obtain the displacements of the cement plug 12 over the course of time t and to obtain the stresses σ and the pore pressures P_(p) experienced by the cement plug 12. Furthermore, this model makes it possible to obtain the evolution of the micro-ring fracture 25 over the course of time t.

Preferably, this calculation is performed before the calculation of the hydration pitch in order to verify whether the micro-ring fracture 25 is open. Indeed, the application of the axial pressure induces a potential closing of the micro-ring fracture 25 due to the Poisson effect.

Thus, when a micro-ring fracture 25 opens between the cement plug and the walls of the well 10 as illustrated in FIG. 3, head losses occur due to the plastic threshold of the drilling mud, which decreases the value of the pressure in the micro-ring fracture proportionally.

Assuming that the flow rates are negligible, the head losses are written as follows:

${\delta \; p_{f}} = \frac{2{\tau_{y} \cdot \delta}\; z}{e}$

wherein e corresponds to the opening of the micro-ring fracture 25, τ_(y) is the plastic threshold of the drilling mud and δz is the depth increment calculated along the trajectory of the well 10. Thus, for a depth interval, the head losses are evaluated by means of the following integral:

${\Delta \; p_{f}} = {2{\int_{z_{1}}^{z_{2}}{\frac{\tau_{y}}{e}{dz}}}}$

As a result, the pressure at the depth z₂ can be calculated from that at the depth z₁ using the following equation:

$P_{mud}^{2} = {P_{mud}^{1} + {{\rho \cdot g \cdot \cos}\; {{incl} \cdot \left( {z_{2} - z_{1}} \right)}} - {2{\int_{z_{1}}^{z_{2}}{\frac{\tau_{y}}{e}{dz}}}}}$

where the second and third terms to the right of the equal sign characterize the pressure increase due to the density of the drilling mud and the decrease in the pressure due to the friction in relation to the plastic threshold of the drilling mud.

A new hypothesis is considered according to which the micro-ring fracture 25 is a linear function of the depth for the considered depth interval, for a digital geometric increment, according to the following equation:

e=A·z+B

wherein A and B are written:

$A = \frac{e_{1} - e_{2}}{z_{1} - z_{2}}$ $B = \frac{{z_{1} \cdot e_{2}} - {z_{2} \cdot e_{1}}}{z_{1} - z_{2}}$

According to another hypothesis, the opening of the micro-ring fracture 25 is a linear function of the pressure according to the following formula:

P _(mud) =K·e+P ₀

where K and P₀ are determined constants.

The equation to be solved in order to evaluate the opening e₂ of the micro-ring fracture at the depth z₂ as a function of the opening of the micro-ring fracture e₁ at the depth z₁ propagating downward is therefore written:

a·x ² +b·x+c·ln x+d=0

with:

x=e ₂

a=K ₂

b=P ₀₂ −P ₀₁−(K ₁ +K ₂)·(e ₁)−ρ·g·cos incl·(z ₂ −z ₁)

c=2·τ_(y)·(z ₂ −z ₁)

d=[K ₁ ·e ₁ +P ₀₁ −P ₀₂ +ρ·g cos incl·(z ₂ −z ₁)]·e ₁−2τ_(y)·(z ₂ −z ₁)·ln e ₁

This equation makes it possible to calculate the pressure losses in the micro-ring fracture 25 assuming a nil flow rate.

If there is flow within the micro-ring fracture Q, this modifies the head losses. Assuming a Herschel-Bulkley fluid and the Poiseuille approximation, the flow, in a steady state, in a micro-ring fracture 25 can be calculated from the following equations:

$\frac{n \cdot \left( {1 - \psi} \right)^{1 + \frac{1}{n}} \cdot \left( {1 + n + {n\; \psi}} \right)}{\left( {n + 1} \right) \cdot \left( {{2n} + 1} \right) \cdot \psi^{\frac{1}{n}}} = {\frac{16Q}{\pi \cdot \left( {d_{outer} - d_{inner}} \right)^{2} \cdot \left( {d_{outer} + d_{inner}} \right)} \cdot \left( \frac{k}{\tau_{y}} \right)^{\frac{1}{n}}}$ $\mspace{20mu} {{\delta \; p_{f}} = {\frac{1}{\psi} \cdot \frac{4\; {\tau_{y} \cdot \delta}\; z}{d_{outer} - d_{inner}}}}$

wherein d_(outer) and d_(inner) are the concave side and convex side diameters of the micro-ring fracture 25.

Assuming that the opening of the micro-ring fracture 25 is small compared to the diameter of the plug according to the following equations:

d _(outer) −d _(inner)=2e

d _(outer) +d _(inner)=4r ₀

Assuming that the term ψ is a linear function of the depth for the considered depth interval according to the following equation:

ψ=A′·z+B′

wherein:

$A^{\prime} = \frac{\psi_{1} - \psi_{2}}{z_{1} - z_{2}}$ $B^{\prime} = \frac{{z_{1} \cdot \psi_{2}} - {z_{2} \cdot \psi_{1}}}{z_{1} - z_{2}}$

The system of equations to be solved in order to evaluate the opening e₂ and the parameter ψ₂ at the depth z₂ as a function of the opening e₁ and the parameter ψ₁ at the depth z₁ for a micro-ring fracture 25 propagating upward is therefore written:

${{K_{2} \cdot e_{2}} + P_{02}} = {{K_{1} \cdot e_{1}} + P_{01} - {{\rho \cdot g \cdot \cos}\; {{incl} \cdot \left( {z_{2} - z_{1}} \right)}} + {\frac{2{\tau_{y} \cdot \left( {z_{1} - z_{2}} \right)}}{{\psi_{1} \cdot e_{2}} - {\psi_{2} \cdot e_{1}}} \cdot \left( {{\ln \; \frac{e_{2}}{\psi_{2}}} - {\ln \; \frac{e_{1}}{\psi_{1}}}} \right)}}$ $\mspace{20mu} {\frac{3{n \cdot \left( {1 - \psi_{2}} \right)^{1 + \frac{1}{n}} \cdot \left( {1 + n + {n\; \psi_{2}}} \right)}}{\left( {n + 1} \right) \cdot \left( {{2n} + 1} \right) \cdot \psi_{2}^{\frac{1}{n}}} = {\frac{3Q}{\pi \cdot e_{2}^{2} \cdot r_{0}} \cdot \left( \frac{k}{\tau_{y}} \right)^{\frac{1}{n\;}}}}$

Solving these equations as a function of the properties of the cement therefore makes it possible to modify the flow rate of the micro-ring fracture 25. Preferably, the cement plug 12 should not have any faults. Furthermore, a fault may be tolerated at the cement plug 12 if the flow rate of the micro-ring fracture 25 is very low. Conversely, if the flow rate of the micro-ring fracture 25 is above a threshold value, it is necessary to redesign the cement plug 12.

One method for redesigning the cement plug 12 consists in determining the quantity of post-expansion agents necessary in order to increase the volume of the cement plug 12 such as to fill in the determined dimensions of the micro-ring fracture 25, as described in step 106 of the method.

Indeed, the cement tends to decrease in volume during the hydration thereof, insofar as the volume of the products is smaller than that of the reactants. However, there are many additives making it possible to result in an increase in volume of the cement. Some are based upon mixtures of calcium sulfates or sodium sulfate or plaster in order to generate Ettringite, for example according to the following formula:

$\left. {{6{Ca}^{2 +}} + {2{{Al}({OH})}_{4}^{-}} + {4{OH}^{-}} + {3{SO}_{4}^{2 -}} + {26H_{2}O}}\Leftrightarrow\underset{\underset{Ettringite}{}}{{{{Ca}_{6}\left\lbrack {{Al}({OH})}_{6} \right\rbrack}_{2} \cdot \left( {SO}_{4} \right)_{2} \cdot 26}H_{2}O} \right.$

Others are based upon compositions of calcined metal oxides, such as magnesium oxide or calcium oxide according to the following formulas:

$\left. {{MgO} + {H_{2}O}}\Leftrightarrow\underset{\underset{Brucite}{}}{{{Mg}({OH})}_{2}} \right.$ $\left. {{CaO} + {H_{2}O}}\Leftrightarrow\underset{\underset{{Calcium}\mspace{14mu} {hydroxide}}{}}{{{Ca}({OH})}_{2}} \right.$

Many factors affect the expansion of cement systems containing such additives, including the type of additive, the additive concentration, the pressure, the temperature. The mineral produced tends to precipitate within those areas where the stresses are lowest, i.e., in the pores or empty spaces of the cement system, causing an expansion of the cement, which is measured in the laboratory, and which may reach several percent.

The chemical reactions relating to the post-expansion may be simulated in the same way as the hydration of the cement, but with different constants. Therefore, the models described in the preceding parts apply identically.

Another manner of simulating them consists in only assigning them a variation in volume, without altering the other properties of the cement. As a result, this makes the problem identical to that of thermal expansion.

Furthermore, the presence of post-expansion agents may decrease the mechanical strength of the cement plug 12. In all cases, i.e., with or without post-expansion agents, the method may also verify, over the course of time t, whether the stresses experienced by the cement plug 12 risk resulting in damage.

For example, the Mohr-Coulomb criterion may calculate the risk of shear damage according to the following equations:

σ₁ − q ⋅ σ₃ − (1 − q) ⋅ p_(p) − UCS ≤ 0 ${UCS} = \frac{2c\; \cos \; \phi}{1 - {\sin \; \phi}}$ $q = \frac{1 + {\sin \; \phi}}{1 - {\sin \; \phi}}$

Furthermore, the risk of tensile damage may be evaluated according to the following equation:

σ₃ −p _(p) +TS≥0

the variables σ₁ and σ₃ representing the major and minor primary stresses.

These criteria make it possible to guarantee the integrity of the cement plug 12 in shear and traction. Alternatively, other criteria can be applied.

Furthermore, the modeling of the cement plug 12 must be concluded, for example if no micro-ring fracture 25 forms.

In practice, this modeling is performed until an end of setting is detected, performed in step 110 of the method. For example, the detection of an end of setting occurs when the evolution of the degree of hydration ξ per hour is less than 2·10⁻⁴. Alternatively, the modeling can be done over a predetermined time interval.

Although no micro-ring fracture 25 has been detected up to the end of setting, conditions exist under which a chance micro-ring fracture 25 may appear on the cement plug 12. In order to verify the strength of the cement plug 12 after the appearance of a chance micro-ring fracture 25, it is possible to determine, in step 111, the pressures Pinf and Psup experienced by the cement plug 12 on the ends 20-21 thereof after setting. These pressures Pinf and Psusp make it possible to determine, in step 112, the stresses experienced by the cement plug 12 after setting and to simulate the propagation of a chance micro-ring fracture 25.

As before, if the chance micro-ring fracture 25 extends between two critical points A-D, there is cause to determine whether it is necessary to correct the characteristics of the cement plug 12, for example if the flow rate of the chance micro-ring fracture 25 is above a threshold value.

The invention thus allows for precise modeling of a cement plug 12 during the hydration thereof, and even after the hydration thereof, in order to determine whether the cement plug 12 is correctly dimensioned in order to meet the constraints of the environment thereof. Furthermore, the invention also makes it possible to precisely correct the characteristics of the cement plug 12, for example by determining the quantity of post-expansion agents based upon the dimensions of the detected micro-ring fracture 25. 

1. A method for determining the integrity of a plug closing an oil well, the method comprising the following steps: digitally modeling at least one structural characteristic of the plug, over the course of time, as the plug solidifies under depth conditions; detecting, over the course of time, the formation of a micro-ring fracture and/or the deterioration of the plug as a function of the at least one structural characteristic of the plug with respect to a micro-ring fracture formation and/or rupture criterion; when the formation of a micro-ring fracture is detected, digitally modeling at least one structural characteristic of the plug, over the course of time, as the plug solidifies under depth conditions on the basis of the dimensional evolution of the micro-ring fracture over the course of time; and if the micro-ring fracture extends between two critical points of the well and/or if the plug is damaged, detecting a fault within the plug.
 2. The determining method according to claim 1, wherein the step for digitally modeling at least one structural characteristic of the plug, over the course of time, as the plug solidifies under depth conditions is performed based upon: properties of the plug; geometric characteristics of the plug; properties of the well; and environmental conditions.
 3. The determining method according to claim 1, wherein the step for digitally modeling at least one structural characteristic of the plug, over the course of time, as the plug solidifies under depth conditions takes account of the evolution of the stresses experienced by the plug, the evolution of the deformation of the plug, the evolution of the chemical setting reaction of the plug, the evolution of the temperature at the plug during setting, the evolution of the properties of the plug and the evolution of the pressure of the pores of the plug.
 4. The determining method according to claim 1, wherein the method also includes the following additional steps: determining the dimensions of the micro-ring fracture; and determining the quantity of post-expansion agents necessary to increase the volume of the plug in order to fill in the determined dimensions of the micro-ring fracture.
 5. The determining method according to claim 1, wherein the tensile damage of the plug is detected.
 6. The determining method according to claim 1, wherein the shear damage of the plug is detected, for example according to the Mohr-Coulomb criterion.
 7. The determining method according to claim 1, wherein, during the step for digitally modeling at least one structural characteristic of the plug, over the course of time, as the plug solidifies under depth conditions, the method also includes the following steps: detecting an end of setting corresponding to an evolution of the degree of solidification per hour of the plug below 2·10⁻⁴; determining a pressure applied above the plug from fluids placed above the plug after detecting an end of setting; determining a pressure applied below the plug from fluids placed below the plug after detecting an end of setting; and determining the stresses experienced by the plug, after setting, based upon pressures applied above and below the plug.
 8. The determining method according to claim 7, wherein after the step consisting of determining the stresses experienced by the plug, after setting, as a function of the pressures applied above and below the plug, the method also includes the following additional steps: simulating the propagation of a chance micro-ring fracture that appears above or below the plug, as a function of the pressures applied above and below the plug; if the chance micro-ring fracture extends between two critical points, determining a flow rate of the chance micro-ring fracture; and if the flow rate of the chance micro-ring fracture is above a threshold, detecting a fault with the plug.
 9. The determining method according to claim 8, wherein after the step consisting of detecting a fault on the plug by the presence of a chance micro-ring fracture, the method also comprises the following steps: determining the dimension of the chance micro-ring fracture; and determining the quantity of post-expansion agents necessary in order to increase the volume of the plug in order to fill in the determined dimension of the micro-ring fracture. 